I agree with most of things posted here, although I must object to the notion, that English would be simpler without the Latin and the French influencing it.
English is simpler than Icelandic by far, it is simpler than German (in my opinion), and it is equally complicated as Danish, (again in my opinion).
Our chaotic spelling situation might be simpler, though - which is what people are often thinking of when they say that English is complex/hard to learn. Apart from a few odd-to-us specifics, such as the use of
eth and
thorn for soft and hard "th", and "gg" instead of "dg" (brigge = bridge), etc., Old English spelling was fairly straightforward and consistant - more or less, English written using the conventions of Vulgar Latin.
The prospective evolution of the English language itself usually gets slighted in alternate-1066 TLs. The grammatical changes, such as the decay of the inflection system and grammatical gender, were already ongoing before the Conquest (as shown by "mistakes" in later OE texts), and probably would have continued. The conservative influence of formal education might slow them down a bit, but more likely writing would merely lag behind the spoken language.
A victory by Hardrada, if consolidated across England, might reinforce Northern pronunciations, e.g., dike instead of ditch, skirt instead of shirt - rather than both forms surviving with variant meanings.
A Norman defeat would have a dramatic effect on vocabulary - farewell to all those French words. (Though if, as I suspect, later medieval England ends up in the French cultural orbit as much of Western Europe did, a fair number of French words may still come in later.)
The end result, by the TL's 20th century, might well be an English that we could understand and even speak fairly well - so long as we stuck to a 5-year-old's vocabulary. Sophisticated concepts ... well, I don't know the native English words for "sophisticated concepts," but it surely had them, because Anglo-Saxon England had been the intellectual center of Europe.
We might be able to half-understand a TV sitcom. The TV news would be "seventeen more gibberish were found dead today, as gibberish gibberish turned into open fighting ..." The newspapers would be totally incomprehensible.
-- Rick